A Stochastic Two-Point Boundary Value Problem

Abstract

We investigate the two-point stochastic boundary-value problem on $[0,1]$: \begin{equation}\label{1} \begin{split} U'' &= f(U)\dot W + g(U,U')\\ U(0) &= \xi\\ U(1)&= \eta. \end{split} \tag{1} \end{equation} where $\dot W$ is a white noise on $[0,1]$, $\xi$ and $\eta$ are random variables, and $f$ and $g$ are continuous real-valued functions. This is the stochastic analogue of the deterministic two point boundary-value problem, which is a classical example of bifurcation. We find that if $f$ and $g$ are affine, there is no bifurcation: for any r.v. $\xi$ and $\eta$, (1) has a unique solution a.s. However, as soon as $f$ is non-linear, bifurcation appears. We investigate the question of when there is either no solution whatsoever, a unique solution, or multiple solutions. We give examples to show that all these possibilities can arise. While our results involve conditions on $f$ and $g$, we conjecture that the only case in which there is no bifurcation is when $f$ is affine.